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Daily Math Minute

Exponents & Polynomials

Multiplying Polynomials

Multiplying binomials and polynomials using the distributive property.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Adding & Subtracting Polynomials

FOIL Is Just Distributing Twice

Multiplying (x + 3)(x + 5) means distributing the entire first binomial across the second — but before reading on, try expanding it by treating (x + 3) as a single number multiplying (x + 5) first, distributing that way, and see how many total terms come out.

Distributing (x + 3) across (x + 5) gives x(x + 5) + 3(x + 5). Distributing each of those pieces individually gives x² + 5x + 3x + 15, which combines to x² + 8x + 15. FOIL — First, Outer, Inner, Last — is just a memorable shortcut for tracking those exact four products (x·x, x·5, 3·x, 3·5) without writing out the full two-step distribution every time.

Worked Example — Multiplying Two Binomials with FOIL

Multiply (2x − 1)(x + 4). First: 2x · x = 2x². Outer: 2x · 4 = 8x. Inner: −1 · x = −x. Last: −1 · 4 = −4. Combine: 2x² + 8x − x − 4 = 2x² + 7x − 4.

Worked Example — Multiplying a Binomial by a Trinomial

Multiply (x + 2)(x² + 3x − 5). Distribute every term of the binomial across every term of the trinomial: x(x² + 3x − 5) + 2(x² + 3x − 5) = x³ + 3x² − 5x + 2x² + 6x − 10. Combine like terms: x³ + 5x² + x − 10.

Worked Example — Squaring a Binomial

Expand (x + 4)². This means (x + 4)(x + 4). Using FOIL: x² + 4x + 4x + 16 = x² + 8x + 16 — notice the middle term is double the product of x and 4, a pattern worth remembering.

Tip

For any polynomial multiplication beyond two binomials, skip trying to remember FOIL and use the general rule directly: distribute every term of the first polynomial across every term of the second, then combine like terms.

Common Mistakes

  • Forgetting one of the four FOIL products, most often the 'Outer' or 'Inner' term.

    Write out all four products explicitly — First, Outer, Inner, Last — before combining anything, rather than trying to combine terms while still multiplying.

  • Squaring a binomial by squaring each term separately, such as expanding (x + 4)² as x² + 16, skipping the middle term entirely.

    (x + 4)² means (x + 4)(x + 4), which must be fully expanded with FOIL or the distributive property — it always produces a middle term (here, 8x) that a shortcut of squaring each piece separately would miss.

Key Takeaways

  • FOIL is a memory device for the four products created when distributing one binomial across another.
  • Multiplying larger polynomials uses the same idea: distribute every term of the first across every term of the second, then combine like terms.
  • Squaring a binomial always produces three terms, including a middle term that's easy to accidentally skip.

Summary

Multiplying polynomials extends the distributive property to expressions with several terms on each side. The next unit works in the opposite direction — factoring — starting with pulling out a shared common factor.

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